LMFDB - Embedded newform 4212.2.i.j.2809.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4212,2,Mod(1405,4212)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4212, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4212.1405"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 4212 = 2^{2} \cdot 3^{4} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4212.i (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,3,0,-2,0,0,0,0,0,-1,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$33.6329893314$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1404)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			2809.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	4212.2809
Dual form			4212.2.i.j.1405.1

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.50000 + 2.59808i) q^{5} +(-1.00000 + 1.73205i) q^{7} +(-0.500000 - 0.866025i) q^{13} +2.00000 q^{19} +(3.00000 + 5.19615i) q^{23} +(-2.00000 + 3.46410i) q^{25} +(-3.00000 + 5.19615i) q^{29} +(-1.00000 - 1.73205i) q^{31} -6.00000 q^{35} +2.00000 q^{37} +(-3.00000 - 5.19615i) q^{41} +(-2.50000 + 4.33013i) q^{43} +(-1.50000 + 2.59808i) q^{47} +(1.50000 + 2.59808i) q^{49} +12.0000 q^{53} +(1.50000 + 2.59808i) q^{59} +(-5.50000 + 9.52628i) q^{61} +(1.50000 - 2.59808i) q^{65} +(-4.00000 - 6.92820i) q^{67} +9.00000 q^{71} -4.00000 q^{73} +(-4.00000 + 6.92820i) q^{79} +(-4.50000 + 7.79423i) q^{83} -3.00000 q^{89} +2.00000 q^{91} +(3.00000 + 5.19615i) q^{95} +(-4.00000 + 6.92820i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 3 q^{5} - 2 q^{7} - q^{13} + 4 q^{19} + 6 q^{23} - 4 q^{25} - 6 q^{29} - 2 q^{31} - 12 q^{35} + 4 q^{37} - 6 q^{41} - 5 q^{43} - 3 q^{47} + 3 q^{49} + 24 q^{53} + 3 q^{59} - 11 q^{61} + 3 q^{65}+ \cdots - 8 q^{97}+O(q^{100}) $ 2 * q + 3 * q^5 - 2 * q^7 - q^13 + 4 * q^19 + 6 * q^23 - 4 * q^25 - 6 * q^29 - 2 * q^31 - 12 * q^35 + 4 * q^37 - 6 * q^41 - 5 * q^43 - 3 * q^47 + 3 * q^49 + 24 * q^53 + 3 * q^59 - 11 * q^61 + 3 * q^65 - 8 * q^67 + 18 * q^71 - 8 * q^73 - 8 * q^79 - 9 * q^83 - 6 * q^89 + 4 * q^91 + 6 * q^95 - 8 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4212\mathbb{Z}\right)^\times$.

$n$	$2107$	$3485$	$3889$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	1.50000	+	2.59808i	0.670820	+	1.16190i	0.977672	+	0.210138i	$0.0673912\pi$
$5$	1.50000	+	2.59808i	0.670820	+	1.16190i	−0.306851	+	0.951757i	$0.599275\pi$
$6$		0			0
$7$	−1.00000	+	1.73205i	−0.377964	+	0.654654i	−0.990766	−	0.135583i	$-0.956709\pi$
$7$	−1.00000	+	1.73205i	−0.377964	+	0.654654i	0.612801	+	0.790237i	$0.290043\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$11$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$12$		0			0
$13$	−0.500000	−	0.866025i	−0.138675	−	0.240192i
$13$	−0.500000	−	0.866025i	−0.138675	−	0.240192i
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$		0			0
$19$	2.00000			0.458831			0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000			0.458831			0.229416	+	0.973329i	$0.426318\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	3.00000	+	5.19615i	0.625543	+	1.08347i	0.988436	+	0.151642i	$0.0484560\pi$
$23$	3.00000	+	5.19615i	0.625543	+	1.08347i	−0.362892	+	0.931831i	$0.618211\pi$
$24$		0			0
$25$	−2.00000	+	3.46410i	−0.400000	+	0.692820i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−3.00000	+	5.19615i	−0.557086	+	0.964901i	0.440652	+	0.897678i	$0.354747\pi$
$29$	−3.00000	+	5.19615i	−0.557086	+	0.964901i	−0.997738	+	0.0672232i	$0.978586\pi$
$30$		0			0
$31$	−1.00000	−	1.73205i	−0.179605	−	0.311086i	0.762140	−	0.647412i	$-0.224149\pi$
$31$	−1.00000	−	1.73205i	−0.179605	−	0.311086i	−0.941745	+	0.336327i	$0.890815\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−6.00000			−1.01419
$36$		0			0
$37$	2.00000			0.328798			0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000			0.328798			0.164399	+	0.986394i	$0.447432\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−3.00000	−	5.19615i	−0.468521	−	0.811503i	0.530831	−	0.847477i	$-0.321880\pi$
$41$	−3.00000	−	5.19615i	−0.468521	−	0.811503i	−0.999353	+	0.0359748i	$0.988546\pi$
$42$		0			0
$43$	−2.50000	+	4.33013i	−0.381246	+	0.660338i	−0.991241	−	0.132068i	$-0.957838\pi$
$43$	−2.50000	+	4.33013i	−0.381246	+	0.660338i	0.609994	+	0.792406i	$0.291172\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−1.50000	+	2.59808i	−0.218797	+	0.378968i	−0.954441	−	0.298401i	$-0.903547\pi$
$47$	−1.50000	+	2.59808i	−0.218797	+	0.378968i	0.735643	+	0.677369i	$0.236880\pi$
$48$		0			0
$49$	1.50000	+	2.59808i	0.214286	+	0.371154i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	12.0000			1.64833			0.824163	−	0.566352i	$-0.191646\pi$
$53$	12.0000			1.64833			0.824163	+	0.566352i	$0.191646\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	1.50000	+	2.59808i	0.195283	+	0.338241i	0.946993	−	0.321253i	$-0.104104\pi$
$59$	1.50000	+	2.59808i	0.195283	+	0.338241i	−0.751710	+	0.659494i	$0.770771\pi$
$60$		0			0
$61$	−5.50000	+	9.52628i	−0.704203	+	1.21972i	0.262776	+	0.964857i	$0.415362\pi$
$61$	−5.50000	+	9.52628i	−0.704203	+	1.21972i	−0.966978	+	0.254858i	$0.917971\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	1.50000	−	2.59808i	0.186052	−	0.322252i
$66$		0			0
$67$	−4.00000	−	6.92820i	−0.488678	−	0.846415i	0.511237	−	0.859440i	$-0.329187\pi$
$67$	−4.00000	−	6.92820i	−0.488678	−	0.846415i	−0.999915	+	0.0130248i	$0.995854\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	9.00000			1.06810			0.534052	−	0.845452i	$-0.320669\pi$
$71$	9.00000			1.06810			0.534052	+	0.845452i	$0.320669\pi$
$72$		0			0
$73$	−4.00000			−0.468165			−0.234082	−	0.972217i	$-0.575209\pi$
$73$	−4.00000			−0.468165			−0.234082	+	0.972217i	$0.575209\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−4.00000	+	6.92820i	−0.450035	+	0.779484i	−0.998388	−	0.0567635i	$-0.981922\pi$
$79$	−4.00000	+	6.92820i	−0.450035	+	0.779484i	0.548352	+	0.836247i	$0.315255\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−4.50000	+	7.79423i	−0.493939	+	0.855528i	−0.999976	−	0.00698436i	$-0.997777\pi$
$83$	−4.50000	+	7.79423i	−0.493939	+	0.855528i	0.506036	+	0.862512i	$0.331110\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−3.00000			−0.317999			−0.159000	−	0.987279i	$-0.550827\pi$
$89$	−3.00000			−0.317999			−0.159000	+	0.987279i	$0.550827\pi$
$90$		0			0
$91$	2.00000			0.209657
$92$		0			0
$93$		0			0
$94$		0			0
$95$	3.00000	+	5.19615i	0.307794	+	0.533114i
$96$		0			0
$97$	−4.00000	+	6.92820i	−0.406138	+	0.703452i	−0.994453	−	0.105180i	$-0.966458\pi$
$97$	−4.00000	+	6.92820i	−0.406138	+	0.703452i	0.588315	+	0.808632i	$0.299792\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$101$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$102$		0			0
$103$	−5.50000	−	9.52628i	−0.541931	−	0.938652i	−0.998793	−	0.0491146i	$-0.984360\pi$
$103$	−5.50000	−	9.52628i	−0.541931	−	0.938652i	0.456862	−	0.889538i	$-0.348973\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−18.0000			−1.74013			−0.870063	−	0.492941i	$-0.835922\pi$
$107$	−18.0000			−1.74013			−0.870063	+	0.492941i	$0.835922\pi$
$108$		0			0
$109$	−4.00000			−0.383131			−0.191565	−	0.981480i	$-0.561356\pi$
$109$	−4.00000			−0.383131			−0.191565	+	0.981480i	$0.561356\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	3.00000	+	5.19615i	0.282216	+	0.488813i	0.971930	−	0.235269i	$-0.0755971\pi$
$113$	3.00000	+	5.19615i	0.282216	+	0.488813i	−0.689714	+	0.724082i	$0.742264\pi$
$114$		0			0
$115$	−9.00000	+	15.5885i	−0.839254	+	1.45363i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	5.50000	+	9.52628i	0.500000	+	0.866025i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	3.00000			0.268328
$126$		0			0
$127$	−13.0000			−1.15356			−0.576782	−	0.816898i	$-0.695692\pi$
$127$	−13.0000			−1.15356			−0.576782	+	0.816898i	$0.695692\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−6.00000	−	10.3923i	−0.524222	−	0.907980i	−0.999602	−	0.0281993i	$-0.991023\pi$
$131$	−6.00000	−	10.3923i	−0.524222	−	0.907980i	0.475380	−	0.879781i	$-0.342311\pi$
$132$		0			0
$133$	−2.00000	+	3.46410i	−0.173422	+	0.300376i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	9.00000	−	15.5885i	0.768922	−	1.33181i	−0.169226	−	0.985577i	$-0.554127\pi$
$137$	9.00000	−	15.5885i	0.768922	−	1.33181i	0.938148	−	0.346235i	$-0.112540\pi$
$138$		0			0
$139$	−5.50000	−	9.52628i	−0.466504	−	0.808008i	0.532764	−	0.846264i	$-0.321153\pi$
$139$	−5.50000	−	9.52628i	−0.466504	−	0.808008i	−0.999268	+	0.0382553i	$0.987820\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	−18.0000			−1.49482
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−7.50000	−	12.9904i	−0.614424	−	1.06421i	−0.990485	−	0.137619i	$-0.956055\pi$
$149$	−7.50000	−	12.9904i	−0.614424	−	1.06421i	0.376061	−	0.926595i	$-0.377278\pi$
$150$		0			0
$151$	−4.00000	+	6.92820i	−0.325515	+	0.563809i	−0.981617	−	0.190864i	$-0.938871\pi$
$151$	−4.00000	+	6.92820i	−0.325515	+	0.563809i	0.656101	+	0.754673i	$0.272204\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	3.00000	−	5.19615i	0.240966	−	0.417365i
$156$		0			0
$157$	3.50000	+	6.06218i	0.279330	+	0.483814i	0.971219	−	0.238190i	$-0.0765542\pi$
$157$	3.50000	+	6.06218i	0.279330	+	0.483814i	−0.691888	+	0.722005i	$0.743221\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−12.0000			−0.945732
$162$		0			0
$163$	−4.00000			−0.313304			−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000			−0.313304			−0.156652	+	0.987654i	$0.550070\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−4.50000	−	7.79423i	−0.348220	−	0.603136i	0.637713	−	0.770274i	$-0.279881\pi$
$167$	−4.50000	−	7.79423i	−0.348220	−	0.603136i	−0.985933	+	0.167139i	$0.946547\pi$
$168$		0			0
$169$	−0.500000	+	0.866025i	−0.0384615	+	0.0666173i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	12.0000	−	20.7846i	0.912343	−	1.58022i	0.101598	−	0.994826i	$-0.467605\pi$
$173$	12.0000	−	20.7846i	0.912343	−	1.58022i	0.810745	−	0.585399i	$-0.199062\pi$
$174$		0			0
$175$	−4.00000	−	6.92820i	−0.302372	−	0.523723i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−6.00000			−0.448461			−0.224231	−	0.974536i	$-0.571987\pi$
$179$	−6.00000			−0.448461			−0.224231	+	0.974536i	$0.571987\pi$
$180$		0			0
$181$	−7.00000			−0.520306			−0.260153	−	0.965567i	$-0.583773\pi$
$181$	−7.00000			−0.520306			−0.260153	+	0.965567i	$0.583773\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	3.00000	+	5.19615i	0.220564	+	0.382029i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	6.00000	−	10.3923i	0.434145	−	0.751961i	−0.563081	−	0.826402i	$-0.690384\pi$
$191$	6.00000	−	10.3923i	0.434145	−	0.751961i	0.997225	+	0.0744412i	$0.0237173\pi$
$192$		0			0
$193$	8.00000	+	13.8564i	0.575853	+	0.997406i	0.995948	+	0.0899262i	$0.0286631\pi$
$193$	8.00000	+	13.8564i	0.575853	+	0.997406i	−0.420096	+	0.907480i	$0.638004\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	21.0000			1.49619			0.748094	−	0.663593i	$-0.230969\pi$
$197$	21.0000			1.49619			0.748094	+	0.663593i	$0.230969\pi$
$198$		0			0
$199$	−25.0000			−1.77220			−0.886102	−	0.463491i	$-0.846597\pi$
$199$	−25.0000			−1.77220			−0.886102	+	0.463491i	$0.846597\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−6.00000	−	10.3923i	−0.421117	−	0.729397i
$204$		0			0
$205$	9.00000	−	15.5885i	0.628587	−	1.08875i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−2.50000	−	4.33013i	−0.172107	−	0.298098i	0.767049	−	0.641588i	$-0.221724\pi$
$211$	−2.50000	−	4.33013i	−0.172107	−	0.298098i	−0.939156	+	0.343490i	$0.888391\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−15.0000			−1.02299
$216$		0			0
$217$	4.00000			0.271538
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−4.00000	+	6.92820i	−0.267860	+	0.463947i	−0.968309	−	0.249756i	$-0.919650\pi$
$223$	−4.00000	+	6.92820i	−0.267860	+	0.463947i	0.700449	+	0.713702i	$0.252983\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−13.5000	+	23.3827i	−0.896026	+	1.55196i	−0.0634974	+	0.997982i	$0.520225\pi$
$227$	−13.5000	+	23.3827i	−0.896026	+	1.55196i	−0.832529	+	0.553981i	$0.813108\pi$
$228$		0			0
$229$	11.0000	+	19.0526i	0.726900	+	1.25903i	0.958187	+	0.286143i	$0.0923732\pi$
$229$	11.0000	+	19.0526i	0.726900	+	1.25903i	−0.231287	+	0.972886i	$0.574293\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	12.0000			0.786146			0.393073	−	0.919507i	$-0.371412\pi$
$233$	12.0000			0.786146			0.393073	+	0.919507i	$0.371412\pi$
$234$		0			0
$235$	−9.00000			−0.587095
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$239$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$240$		0			0
$241$	−4.00000	+	6.92820i	−0.257663	+	0.446285i	−0.965615	−	0.259975i	$-0.916286\pi$
$241$	−4.00000	+	6.92820i	−0.257663	+	0.446285i	0.707953	+	0.706260i	$0.249619\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−4.50000	+	7.79423i	−0.287494	+	0.497955i
$246$		0			0
$247$	−1.00000	−	1.73205i	−0.0636285	−	0.110208i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	18.0000			1.13615			0.568075	−	0.822977i	$-0.307688\pi$
$251$	18.0000			1.13615			0.568075	+	0.822977i	$0.307688\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−3.00000	−	5.19615i	−0.187135	−	0.324127i	0.757159	−	0.653231i	$-0.226587\pi$
$257$	−3.00000	−	5.19615i	−0.187135	−	0.324127i	−0.944294	+	0.329104i	$0.893253\pi$
$258$		0			0
$259$	−2.00000	+	3.46410i	−0.124274	+	0.215249i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−12.0000	+	20.7846i	−0.739952	+	1.28163i	0.212565	+	0.977147i	$0.431818\pi$
$263$	−12.0000	+	20.7846i	−0.739952	+	1.28163i	−0.952517	+	0.304487i	$0.901515\pi$
$264$		0			0
$265$	18.0000	+	31.1769i	1.10573	+	1.91518i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−6.00000			−0.365826			−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000			−0.365826			−0.182913	+	0.983129i	$0.558553\pi$
$270$		0			0
$271$	8.00000			0.485965			0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000			0.485965			0.242983	+	0.970031i	$0.421874\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	0.500000	−	0.866025i	0.0300421	−	0.0520344i	−0.850613	−	0.525792i	$-0.823769\pi$
$277$	0.500000	−	0.866025i	0.0300421	−	0.0520344i	0.880656	+	0.473757i	$0.157103\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	10.5000	−	18.1865i	0.626377	−	1.08492i	−0.361895	−	0.932219i	$-0.617870\pi$
$281$	10.5000	−	18.1865i	0.626377	−	1.08492i	0.988273	−	0.152699i	$-0.0487965\pi$
$282$		0			0
$283$	3.50000	+	6.06218i	0.208053	+	0.360359i	0.951101	−	0.308879i	$-0.0999539\pi$
$283$	3.50000	+	6.06218i	0.208053	+	0.360359i	−0.743048	+	0.669238i	$0.766621\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	12.0000			0.708338
$288$		0			0
$289$	−17.0000			−1.00000
$290$		0			0
$291$		0			0
$292$		0			0
$293$	3.00000	+	5.19615i	0.175262	+	0.303562i	0.940252	−	0.340480i	$-0.110589\pi$
$293$	3.00000	+	5.19615i	0.175262	+	0.303562i	−0.764990	+	0.644042i	$0.777256\pi$
$294$		0			0
$295$	−4.50000	+	7.79423i	−0.262000	+	0.453798i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	3.00000	−	5.19615i	0.173494	−	0.300501i
$300$		0			0
$301$	−5.00000	−	8.66025i	−0.288195	−	0.499169i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−33.0000			−1.88957
$306$		0			0
$307$	2.00000			0.114146			0.0570730	−	0.998370i	$-0.481823\pi$
$307$	2.00000			0.114146			0.0570730	+	0.998370i	$0.481823\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$311$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$312$		0			0
$313$	15.5000	−	26.8468i	0.876112	−	1.51747i	0.0205381	−	0.999789i	$-0.493462\pi$
$313$	15.5000	−	26.8468i	0.876112	−	1.51747i	0.855574	−	0.517681i	$-0.173205\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−9.00000	+	15.5885i	−0.505490	+	0.875535i	0.494489	+	0.869184i	$0.335355\pi$
$317$	−9.00000	+	15.5885i	−0.505490	+	0.875535i	−0.999980	+	0.00635137i	$0.997978\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	4.00000			0.221880
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−3.00000	−	5.19615i	−0.165395	−	0.286473i
$330$		0			0
$331$	−13.0000	+	22.5167i	−0.714545	+	1.23763i	0.248590	+	0.968609i	$0.420033\pi$
$331$	−13.0000	+	22.5167i	−0.714545	+	1.23763i	−0.963135	+	0.269019i	$0.913301\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	12.0000	−	20.7846i	0.655630	−	1.13558i
$336$		0			0
$337$	3.50000	+	6.06218i	0.190657	+	0.330228i	0.945468	−	0.325714i	$-0.105605\pi$
$337$	3.50000	+	6.06218i	0.190657	+	0.330228i	−0.754811	+	0.655942i	$0.772271\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−20.0000			−1.07990
$344$		0			0
$345$		0			0
$346$		0			0
$347$	3.00000	+	5.19615i	0.161048	+	0.278944i	0.935245	−	0.354001i	$-0.115179\pi$
$347$	3.00000	+	5.19615i	0.161048	+	0.278944i	−0.774197	+	0.632945i	$0.781846\pi$
$348$		0			0
$349$	−7.00000	+	12.1244i	−0.374701	+	0.649002i	−0.990282	−	0.139072i	$-0.955588\pi$
$349$	−7.00000	+	12.1244i	−0.374701	+	0.649002i	0.615581	+	0.788074i	$0.288921\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−7.50000	+	12.9904i	−0.399185	+	0.691408i	−0.993626	−	0.112731i	$-0.964040\pi$
$353$	−7.50000	+	12.9904i	−0.399185	+	0.691408i	0.594441	+	0.804139i	$0.297373\pi$
$354$		0			0
$355$	13.5000	+	23.3827i	0.716506	+	1.24102i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	33.0000			1.74167			0.870837	−	0.491572i	$-0.163578\pi$
$359$	33.0000			1.74167			0.870837	+	0.491572i	$0.163578\pi$
$360$		0			0
$361$	−15.0000			−0.789474
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−6.00000	−	10.3923i	−0.314054	−	0.543958i
$366$		0			0
$367$	8.00000	−	13.8564i	0.417597	−	0.723299i	−0.578101	−	0.815966i	$-0.696206\pi$
$367$	8.00000	−	13.8564i	0.417597	−	0.723299i	0.995697	+	0.0926670i	$0.0295392\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−12.0000	+	20.7846i	−0.623009	+	1.07908i
$372$		0			0
$373$	−2.50000	−	4.33013i	−0.129445	−	0.224205i	0.794017	−	0.607896i	$-0.207986\pi$
$373$	−2.50000	−	4.33013i	−0.129445	−	0.224205i	−0.923462	+	0.383691i	$0.874653\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	6.00000			0.309016
$378$		0			0
$379$	−10.0000			−0.513665			−0.256833	−	0.966456i	$-0.582679\pi$
$379$	−10.0000			−0.513665			−0.256833	+	0.966456i	$0.582679\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	12.0000	+	20.7846i	0.613171	+	1.06204i	0.990702	+	0.136047i	$0.0434398\pi$
$383$	12.0000	+	20.7846i	0.613171	+	1.06204i	−0.377531	+	0.925997i	$0.623227\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	9.00000	−	15.5885i	0.456318	−	0.790366i	−0.542445	−	0.840091i	$-0.682501\pi$
$389$	9.00000	−	15.5885i	0.456318	−	0.790366i	0.998763	+	0.0497253i	$0.0158346\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−24.0000			−1.20757
$396$		0			0
$397$	32.0000			1.60603			0.803017	−	0.595956i	$-0.203227\pi$
$397$	32.0000			1.60603			0.803017	+	0.595956i	$0.203227\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	19.5000	+	33.7750i	0.973784	+	1.68664i	0.683892	+	0.729583i	$0.260286\pi$
$401$	19.5000	+	33.7750i	0.973784	+	1.68664i	0.289891	+	0.957060i	$0.406381\pi$
$402$		0			0
$403$	−1.00000	+	1.73205i	−0.0498135	+	0.0862796i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	5.00000	+	8.66025i	0.247234	+	0.428222i	0.962757	−	0.270367i	$-0.0871450\pi$
$409$	5.00000	+	8.66025i	0.247234	+	0.428222i	−0.715523	+	0.698589i	$0.753812\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−6.00000			−0.295241
$414$		0			0
$415$	−27.0000			−1.32538
$416$		0			0
$417$		0			0
$418$		0			0
$419$	6.00000	+	10.3923i	0.293119	+	0.507697i	0.974546	−	0.224189i	$-0.0719734\pi$
$419$	6.00000	+	10.3923i	0.293119	+	0.507697i	−0.681426	+	0.731887i	$0.738640\pi$
$420$		0			0
$421$	17.0000	−	29.4449i	0.828529	−	1.43505i	−0.0706626	−	0.997500i	$-0.522511\pi$
$421$	17.0000	−	29.4449i	0.828529	−	1.43505i	0.899192	−	0.437555i	$-0.144155\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−11.0000	−	19.0526i	−0.532327	−	0.922018i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−27.0000			−1.30054			−0.650272	−	0.759701i	$-0.725345\pi$
$431$	−27.0000			−1.30054			−0.650272	+	0.759701i	$0.725345\pi$
$432$		0			0
$433$	−22.0000			−1.05725			−0.528626	−	0.848855i	$-0.677293\pi$
$433$	−22.0000			−1.05725			−0.528626	+	0.848855i	$0.677293\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	6.00000	+	10.3923i	0.287019	+	0.497131i
$438$		0			0
$439$	0.500000	−	0.866025i	0.0238637	−	0.0413331i	−0.853847	−	0.520524i	$-0.825737\pi$
$439$	0.500000	−	0.866025i	0.0238637	−	0.0413331i	0.877711	+	0.479191i	$0.159070\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	15.0000	−	25.9808i	0.712672	−	1.23438i	−0.251179	−	0.967941i	$-0.580818\pi$
$443$	15.0000	−	25.9808i	0.712672	−	1.23438i	0.963851	−	0.266443i	$-0.0858483\pi$
$444$		0			0
$445$	−4.50000	−	7.79423i	−0.213320	−	0.369482i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	21.0000			0.991051			0.495526	−	0.868593i	$-0.334975\pi$
$449$	21.0000			0.991051			0.495526	+	0.868593i	$0.334975\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$	3.00000	+	5.19615i	0.140642	+	0.243599i
$456$		0			0
$457$	8.00000	−	13.8564i	0.374224	−	0.648175i	−0.615986	−	0.787757i	$-0.711242\pi$
$457$	8.00000	−	13.8564i	0.374224	−	0.648175i	0.990211	+	0.139581i	$0.0445757\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−16.5000	+	28.5788i	−0.768482	+	1.33105i	0.169904	+	0.985461i	$0.445654\pi$
$461$	−16.5000	+	28.5788i	−0.768482	+	1.33105i	−0.938386	+	0.345589i	$0.887679\pi$
$462$		0			0
$463$	2.00000	+	3.46410i	0.0929479	+	0.160990i	0.908750	−	0.417340i	$-0.137038\pi$
$463$	2.00000	+	3.46410i	0.0929479	+	0.160990i	−0.815802	+	0.578331i	$0.803704\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0			−	1.00000i	$-0.5\pi$
$467$		0			0				1.00000i	$0.5\pi$
$468$		0			0
$469$	16.0000			0.738811
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−4.00000	+	6.92820i	−0.183533	+	0.317888i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−10.5000	+	18.1865i	−0.479757	+	0.830964i	−0.999730	−	0.0232187i	$-0.992609\pi$
$479$	−10.5000	+	18.1865i	−0.479757	+	0.830964i	0.519973	+	0.854183i	$0.325942\pi$
$480$		0			0
$481$	−1.00000	−	1.73205i	−0.0455961	−	0.0789747i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−24.0000			−1.08978
$486$		0			0
$487$	38.0000			1.72194			0.860972	−	0.508652i	$-0.169856\pi$
$487$	38.0000			1.72194			0.860972	+	0.508652i	$0.169856\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$491$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−9.00000	+	15.5885i	−0.403705	+	0.699238i
$498$		0			0
$499$	20.0000	+	34.6410i	0.895323	+	1.55074i	0.833404	+	0.552664i	$0.186389\pi$
$499$	20.0000	+	34.6410i	0.895323	+	1.55074i	0.0619186	+	0.998081i	$0.480278\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	36.0000			1.60516			0.802580	−	0.596544i	$-0.203460\pi$
$503$	36.0000			1.60516			0.802580	+	0.596544i	$0.203460\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	7.50000	+	12.9904i	0.332432	+	0.575789i	0.982988	−	0.183669i	$-0.0587976\pi$
$509$	7.50000	+	12.9904i	0.332432	+	0.575789i	−0.650556	+	0.759458i	$0.725464\pi$
$510$		0			0
$511$	4.00000	−	6.92820i	0.176950	−	0.306486i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	16.5000	−	28.5788i	0.727077	−	1.25933i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	24.0000			1.05146			0.525730	−	0.850652i	$-0.323792\pi$
$521$	24.0000			1.05146			0.525730	+	0.850652i	$0.323792\pi$
$522$		0			0
$523$	−25.0000			−1.09317			−0.546587	−	0.837402i	$-0.684073\pi$
$523$	−25.0000			−1.09317			−0.546587	+	0.837402i	$0.684073\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−6.50000	+	11.2583i	−0.282609	+	0.489493i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−3.00000	+	5.19615i	−0.129944	+	0.225070i
$534$		0			0
$535$	−27.0000	−	46.7654i	−1.16731	−	2.02184i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	44.0000			1.89171			0.945854	−	0.324593i	$-0.105227\pi$
$541$	44.0000			1.89171			0.945854	+	0.324593i	$0.105227\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−6.00000	−	10.3923i	−0.257012	−	0.445157i
$546$		0			0
$547$	8.00000	−	13.8564i	0.342055	−	0.592457i	−0.642759	−	0.766068i	$-0.722210\pi$
$547$	8.00000	−	13.8564i	0.342055	−	0.592457i	0.984814	+	0.173611i	$0.0555436\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−6.00000	+	10.3923i	−0.255609	+	0.442727i
$552$		0			0
$553$	−8.00000	−	13.8564i	−0.340195	−	0.589234i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−18.0000			−0.762684			−0.381342	−	0.924434i	$-0.624538\pi$
$557$	−18.0000			−0.762684			−0.381342	+	0.924434i	$0.624538\pi$
$558$		0			0
$559$	5.00000			0.211477
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−12.0000	−	20.7846i	−0.505740	−	0.875967i	−0.999978	−	0.00664037i	$-0.997886\pi$
$563$	−12.0000	−	20.7846i	−0.505740	−	0.875967i	0.494238	−	0.869326i	$-0.335447\pi$
$564$		0			0
$565$	−9.00000	+	15.5885i	−0.378633	+	0.655811i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−18.0000	+	31.1769i	−0.754599	+	1.30700i	0.190974	+	0.981595i	$0.438835\pi$
$569$	−18.0000	+	31.1769i	−0.754599	+	1.30700i	−0.945573	+	0.325409i	$0.894498\pi$
$570$		0			0
$571$	−11.5000	−	19.9186i	−0.481260	−	0.833567i	0.518509	−	0.855072i	$-0.326487\pi$
$571$	−11.5000	−	19.9186i	−0.481260	−	0.833567i	−0.999769	+	0.0215055i	$0.993154\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−24.0000			−1.00087
$576$		0			0
$577$	2.00000			0.0832611			0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000			0.0832611			0.0416305	+	0.999133i	$0.486745\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−9.00000	−	15.5885i	−0.373383	−	0.646718i
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	12.0000	−	20.7846i	0.495293	−	0.857873i	−0.504692	−	0.863299i	$-0.668394\pi$
$587$	12.0000	−	20.7846i	0.495293	−	0.857873i	0.999985	+	0.00542667i	$0.00172737\pi$
$588$		0			0
$589$	−2.00000	−	3.46410i	−0.0824086	−	0.142736i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	18.0000			0.739171			0.369586	−	0.929197i	$-0.379500\pi$
$593$	18.0000			0.739171			0.369586	+	0.929197i	$0.379500\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	18.0000	+	31.1769i	0.735460	+	1.27385i	0.954521	+	0.298143i	$0.0963673\pi$
$599$	18.0000	+	31.1769i	0.735460	+	1.27385i	−0.219061	+	0.975711i	$0.570299\pi$
$600$		0			0
$601$	−8.50000	+	14.7224i	−0.346722	+	0.600541i	−0.985665	−	0.168714i	$-0.946039\pi$
$601$	−8.50000	+	14.7224i	−0.346722	+	0.600541i	0.638943	+	0.769254i	$0.279372\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−16.5000	+	28.5788i	−0.670820	+	1.16190i
$606$		0			0
$607$	−16.0000	−	27.7128i	−0.649420	−	1.12483i	−0.983262	−	0.182199i	$-0.941678\pi$
$607$	−16.0000	−	27.7128i	−0.649420	−	1.12483i	0.333842	−	0.942629i	$-0.391655\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	3.00000			0.121367
$612$		0			0
$613$	−4.00000			−0.161558			−0.0807792	−	0.996732i	$-0.525741\pi$
$613$	−4.00000			−0.161558			−0.0807792	+	0.996732i	$0.525741\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	4.50000	+	7.79423i	0.181163	+	0.313784i	0.942277	−	0.334835i	$-0.108680\pi$
$617$	4.50000	+	7.79423i	0.181163	+	0.313784i	−0.761114	+	0.648618i	$0.775347\pi$
$618$		0			0
$619$	−16.0000	+	27.7128i	−0.643094	+	1.11387i	0.341644	+	0.939829i	$0.389016\pi$
$619$	−16.0000	+	27.7128i	−0.643094	+	1.11387i	−0.984738	+	0.174042i	$0.944317\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	3.00000	−	5.19615i	0.120192	−	0.208179i
$624$		0			0
$625$	14.5000	+	25.1147i	0.580000	+	1.00459i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	26.0000			1.03504			0.517522	−	0.855670i	$-0.326855\pi$
$631$	26.0000			1.03504			0.517522	+	0.855670i	$0.326855\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−19.5000	−	33.7750i	−0.773834	−	1.34032i
$636$		0			0
$637$	1.50000	−	2.59808i	0.0594322	−	0.102940i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−3.00000	+	5.19615i	−0.118493	+	0.205236i	−0.919171	−	0.393860i	$-0.871140\pi$
$641$	−3.00000	+	5.19615i	−0.118493	+	0.205236i	0.800678	+	0.599095i	$0.204473\pi$
$642$		0			0
$643$	−16.0000	−	27.7128i	−0.630978	−	1.09289i	−0.987352	−	0.158543i	$-0.949320\pi$
$643$	−16.0000	−	27.7128i	−0.630978	−	1.09289i	0.356374	−	0.934344i	$-0.384013\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	30.0000			1.17942			0.589711	−	0.807614i	$-0.299242\pi$
$647$	30.0000			1.17942			0.589711	+	0.807614i	$0.299242\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−12.0000	−	20.7846i	−0.469596	−	0.813365i	0.529799	−	0.848123i	$-0.322267\pi$
$653$	−12.0000	−	20.7846i	−0.469596	−	0.813365i	−0.999396	+	0.0347583i	$0.988934\pi$
$654$		0			0
$655$	18.0000	−	31.1769i	0.703318	−	1.21818i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−3.00000	+	5.19615i	−0.116863	+	0.202413i	−0.918523	−	0.395367i	$-0.870617\pi$
$659$	−3.00000	+	5.19615i	−0.116863	+	0.202413i	0.801660	+	0.597781i	$0.203951\pi$
$660$		0			0
$661$	−10.0000	−	17.3205i	−0.388955	−	0.673690i	0.603354	−	0.797473i	$-0.293830\pi$
$661$	−10.0000	−	17.3205i	−0.388955	−	0.673690i	−0.992309	+	0.123784i	$0.960497\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−12.0000			−0.465340
$666$		0			0
$667$	−36.0000			−1.39393
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	18.5000	−	32.0429i	0.713123	−	1.23516i	−0.250557	−	0.968102i	$-0.580614\pi$
$673$	18.5000	−	32.0429i	0.713123	−	1.23516i	0.963679	−	0.267063i	$-0.0860531\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	24.0000	−	41.5692i	0.922395	−	1.59763i	0.126697	−	0.991941i	$-0.459562\pi$
$677$	24.0000	−	41.5692i	0.922395	−	1.59763i	0.795698	−	0.605693i	$-0.207104\pi$
$678$		0			0
$679$	−8.00000	−	13.8564i	−0.307012	−	0.531760i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−27.0000			−1.03313			−0.516563	−	0.856249i	$-0.672789\pi$
$683$	−27.0000			−1.03313			−0.516563	+	0.856249i	$0.672789\pi$
$684$		0			0
$685$	54.0000			2.06323
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−6.00000	−	10.3923i	−0.228582	−	0.395915i
$690$		0			0
$691$	−7.00000	+	12.1244i	−0.266293	+	0.461232i	−0.967901	−	0.251330i	$-0.919132\pi$
$691$	−7.00000	+	12.1244i	−0.266293	+	0.461232i	0.701609	+	0.712562i	$0.252465\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	16.5000	−	28.5788i	0.625881	−	1.08406i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	24.0000			0.906467			0.453234	−	0.891392i	$-0.350270\pi$
$701$	24.0000			0.906467			0.453234	+	0.891392i	$0.350270\pi$
$702$		0			0
$703$	4.00000			0.150863
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−13.0000	+	22.5167i	−0.488225	+	0.845631i	−0.999908	−	0.0135434i	$-0.995689\pi$
$709$	−13.0000	+	22.5167i	−0.488225	+	0.845631i	0.511683	+	0.859174i	$0.329022\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	6.00000	−	10.3923i	0.224702	−	0.389195i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	48.0000			1.79010			0.895049	−	0.445968i	$-0.147140\pi$
$719$	48.0000			1.79010			0.895049	+	0.445968i	$0.147140\pi$
$720$		0			0
$721$	22.0000			0.819323
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−12.0000	−	20.7846i	−0.445669	−	0.771921i
$726$		0			0
$727$	−4.00000	+	6.92820i	−0.148352	+	0.256953i	−0.930618	−	0.365991i	$-0.880730\pi$
$727$	−4.00000	+	6.92820i	−0.148352	+	0.256953i	0.782267	+	0.622944i	$0.214063\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	26.0000	+	45.0333i	0.960332	+	1.66334i	0.721665	+	0.692242i	$0.243377\pi$
$733$	26.0000	+	45.0333i	0.960332	+	1.66334i	0.238667	+	0.971102i	$0.423290\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	38.0000			1.39785			0.698926	−	0.715194i	$-0.253662\pi$
$739$	38.0000			1.39785			0.698926	+	0.715194i	$0.253662\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−4.50000	−	7.79423i	−0.165089	−	0.285943i	0.771598	−	0.636111i	$-0.219458\pi$
$743$	−4.50000	−	7.79423i	−0.165089	−	0.285943i	−0.936687	+	0.350168i	$0.886124\pi$
$744$		0			0
$745$	22.5000	−	38.9711i	0.824336	−	1.42779i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	18.0000	−	31.1769i	0.657706	−	1.13918i
$750$		0			0
$751$	−16.0000	−	27.7128i	−0.583848	−	1.01125i	−0.995018	−	0.0996961i	$-0.968213\pi$
$751$	−16.0000	−	27.7128i	−0.583848	−	1.01125i	0.411170	−	0.911559i	$-0.365120\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−24.0000			−0.873449
$756$		0			0
$757$	38.0000			1.38113			0.690567	−	0.723269i	$-0.257361\pi$
$757$	38.0000			1.38113			0.690567	+	0.723269i	$0.257361\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−1.50000	−	2.59808i	−0.0543750	−	0.0941802i	0.837557	−	0.546350i	$-0.183983\pi$
$761$	−1.50000	−	2.59808i	−0.0543750	−	0.0941802i	−0.891932	+	0.452170i	$0.850650\pi$
$762$		0			0
$763$	4.00000	−	6.92820i	0.144810	−	0.250818i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	1.50000	−	2.59808i	0.0541619	−	0.0938111i
$768$		0			0
$769$	−7.00000	−	12.1244i	−0.252426	−	0.437215i	0.711767	−	0.702416i	$-0.247895\pi$
$769$	−7.00000	−	12.1244i	−0.252426	−	0.437215i	−0.964193	+	0.265200i	$0.914562\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−45.0000			−1.61854			−0.809269	−	0.587439i	$-0.800136\pi$
$773$	−45.0000			−1.61854			−0.809269	+	0.587439i	$0.800136\pi$
$774$		0			0
$775$	8.00000			0.287368
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−6.00000	−	10.3923i	−0.214972	−	0.372343i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−10.5000	+	18.1865i	−0.374761	+	0.649105i
$786$		0			0
$787$	14.0000	+	24.2487i	0.499046	+	0.864373i	0.999999	−	0.00110111i	$-0.000350496\pi$
$787$	14.0000	+	24.2487i	0.499046	+	0.864373i	−0.500953	+	0.865474i	$0.667017\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−12.0000			−0.426671
$792$		0			0
$793$	11.0000			0.390621
$794$		0			0
$795$		0			0
$796$		0			0
$797$	9.00000	+	15.5885i	0.318796	+	0.552171i	0.980237	−	0.197826i	$-0.0633881\pi$
$797$	9.00000	+	15.5885i	0.318796	+	0.552171i	−0.661441	+	0.749997i	$0.730055\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$	−18.0000	−	31.1769i	−0.634417	−	1.09884i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−36.0000			−1.26569			−0.632846	−	0.774277i	$-0.718114\pi$
$809$	−36.0000			−1.26569			−0.632846	+	0.774277i	$0.718114\pi$
$810$		0			0
$811$	32.0000			1.12367			0.561836	−	0.827249i	$-0.310095\pi$
$811$	32.0000			1.12367			0.561836	+	0.827249i	$0.310095\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−6.00000	−	10.3923i	−0.210171	−	0.364027i
$816$		0			0
$817$	−5.00000	+	8.66025i	−0.174928	+	0.302984i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	3.00000	−	5.19615i	0.104701	−	0.181347i	−0.808915	−	0.587925i	$-0.799945\pi$
$821$	3.00000	−	5.19615i	0.104701	−	0.181347i	0.913616	+	0.406578i	$0.133278\pi$
$822$		0			0
$823$	6.50000	+	11.2583i	0.226576	+	0.392441i	0.956791	−	0.290776i	$-0.0939136\pi$
$823$	6.50000	+	11.2583i	0.226576	+	0.392441i	−0.730215	+	0.683217i	$0.760580\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−12.0000			−0.417281			−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000			−0.417281			−0.208640	+	0.977992i	$0.566904\pi$
$828$		0			0
$829$	11.0000			0.382046			0.191023	−	0.981586i	$-0.438820\pi$
$829$	11.0000			0.382046			0.191023	+	0.981586i	$0.438820\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	13.5000	−	23.3827i	0.467187	−	0.809191i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	10.5000	−	18.1865i	0.362500	−	0.627869i	−0.625871	−	0.779926i	$-0.715257\pi$
$839$	10.5000	−	18.1865i	0.362500	−	0.627869i	0.988372	+	0.152057i	$0.0485899\pi$
$840$		0			0
$841$	−3.50000	−	6.06218i	−0.120690	−	0.209041i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−3.00000			−0.103203
$846$		0			0
$847$	−22.0000			−0.755929
$848$		0			0
$849$		0			0
$850$		0			0
$851$	6.00000	+	10.3923i	0.205677	+	0.356244i
$852$		0			0
$853$	11.0000	−	19.0526i	0.376633	−	0.652347i	−0.613937	−	0.789355i	$-0.710415\pi$
$853$	11.0000	−	19.0526i	0.376633	−	0.652347i	0.990570	+	0.137008i	$0.0437485\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	9.00000	−	15.5885i	0.307434	−	0.532492i	−0.670366	−	0.742030i	$-0.733863\pi$
$857$	9.00000	−	15.5885i	0.307434	−	0.532492i	0.977800	+	0.209539i	$0.0671963\pi$
$858$		0			0
$859$	14.0000	+	24.2487i	0.477674	+	0.827355i	0.999672	−	0.0255910i	$-0.00814674\pi$
$859$	14.0000	+	24.2487i	0.477674	+	0.827355i	−0.521999	+	0.852946i	$0.674813\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−9.00000			−0.306364			−0.153182	−	0.988198i	$-0.548952\pi$
$863$	−9.00000			−0.306364			−0.153182	+	0.988198i	$0.548952\pi$
$864$		0			0
$865$	72.0000			2.44807
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$	−4.00000	+	6.92820i	−0.135535	+	0.234753i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−3.00000	+	5.19615i	−0.101419	+	0.175662i
$876$		0			0
$877$	−16.0000	−	27.7128i	−0.540282	−	0.935795i	−0.998888	−	0.0471555i	$-0.984984\pi$
$877$	−16.0000	−	27.7128i	−0.540282	−	0.935795i	0.458606	−	0.888640i	$-0.348349\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	12.0000			0.404290			0.202145	−	0.979356i	$-0.435209\pi$
$881$	12.0000			0.404290			0.202145	+	0.979356i	$0.435209\pi$
$882$		0			0
$883$	−52.0000			−1.74994			−0.874970	−	0.484178i	$-0.839119\pi$
$883$	−52.0000			−1.74994			−0.874970	+	0.484178i	$0.839119\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	6.00000	+	10.3923i	0.201460	+	0.348939i	0.948999	−	0.315279i	$-0.102098\pi$
$887$	6.00000	+	10.3923i	0.201460	+	0.348939i	−0.747539	+	0.664218i	$0.768765\pi$
$888$		0			0
$889$	13.0000	−	22.5167i	0.436006	−	0.755185i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−3.00000	+	5.19615i	−0.100391	+	0.173883i
$894$		0			0
$895$	−9.00000	−	15.5885i	−0.300837	−	0.521065i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	12.0000			0.400222
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−10.5000	−	18.1865i	−0.349032	−	0.604541i
$906$		0			0
$907$	9.50000	−	16.4545i	0.315442	−	0.546362i	−0.664089	−	0.747653i	$-0.731180\pi$
$907$	9.50000	−	16.4545i	0.315442	−	0.546362i	0.979531	+	0.201291i	$0.0645138\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	18.0000	−	31.1769i	0.596367	−	1.03294i	−0.396986	−	0.917825i	$-0.629944\pi$
$911$	18.0000	−	31.1769i	0.596367	−	1.03294i	0.993352	−	0.115113i	$-0.0367229\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$	24.0000			0.792550
$918$		0			0
$919$	29.0000			0.956622			0.478311	−	0.878191i	$-0.341249\pi$
$919$	29.0000			0.956622			0.478311	+	0.878191i	$0.341249\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−4.50000	−	7.79423i	−0.148119	−	0.256550i
$924$		0			0
$925$	−4.00000	+	6.92820i	−0.131519	+	0.227798i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	15.0000	−	25.9808i	0.492134	−	0.852401i	−0.507825	−	0.861460i	$-0.669550\pi$
$929$	15.0000	−	25.9808i	0.492134	−	0.852401i	0.999959	+	0.00905914i	$0.00288365\pi$
$930$		0			0
$931$	3.00000	+	5.19615i	0.0983210	+	0.170297i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	47.0000			1.53542			0.767712	−	0.640796i	$-0.221395\pi$
$937$	47.0000			1.53542			0.767712	+	0.640796i	$0.221395\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	9.00000	+	15.5885i	0.293392	+	0.508169i	0.974609	−	0.223912i	$-0.0718827\pi$
$941$	9.00000	+	15.5885i	0.293392	+	0.508169i	−0.681218	+	0.732081i	$0.738549\pi$
$942$		0			0
$943$	18.0000	−	31.1769i	0.586161	−	1.01526i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−4.50000	+	7.79423i	−0.146230	+	0.253278i	−0.929831	−	0.367986i	$-0.880047\pi$
$947$	−4.50000	+	7.79423i	−0.146230	+	0.253278i	0.783601	+	0.621264i	$0.213381\pi$
$948$		0			0
$949$	2.00000	+	3.46410i	0.0649227	+	0.112449i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−6.00000			−0.194359			−0.0971795	−	0.995267i	$-0.530982\pi$
$953$	−6.00000			−0.194359			−0.0971795	+	0.995267i	$0.530982\pi$
$954$		0			0
$955$	36.0000			1.16493
$956$		0			0
$957$		0			0
$958$		0			0
$959$	18.0000	+	31.1769i	0.581250	+	1.00676i
$960$		0			0
$961$	13.5000	−	23.3827i	0.435484	−	0.754280i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−24.0000	+	41.5692i	−0.772587	+	1.33816i
$966$		0			0
$967$	2.00000	+	3.46410i	0.0643157	+	0.111398i	0.896390	−	0.443266i	$-0.146180\pi$
$967$	2.00000	+	3.46410i	0.0643157	+	0.111398i	−0.832075	+	0.554664i	$0.812847\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	18.0000			0.577647			0.288824	−	0.957382i	$-0.406736\pi$
$971$	18.0000			0.577647			0.288824	+	0.957382i	$0.406736\pi$
$972$		0			0
$973$	22.0000			0.705288
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1.50000	+	2.59808i	0.0479893	+	0.0831198i	0.889022	−	0.457864i	$-0.151385\pi$
$977$	1.50000	+	2.59808i	0.0479893	+	0.0831198i	−0.841033	+	0.540984i	$0.818052\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−12.0000	+	20.7846i	−0.382741	+	0.662926i	−0.991453	−	0.130465i	$-0.958353\pi$
$983$	−12.0000	+	20.7846i	−0.382741	+	0.662926i	0.608712	+	0.793391i	$0.291686\pi$
$984$		0			0
$985$	31.5000	+	54.5596i	1.00367	+	1.73841i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−30.0000			−0.953945
$990$		0			0
$991$	−25.0000			−0.794151			−0.397076	−	0.917786i	$-0.629975\pi$
$991$	−25.0000			−0.794151			−0.397076	+	0.917786i	$0.629975\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−37.5000	−	64.9519i	−1.18883	−	2.05911i
$996$		0			0
$997$	−17.5000	+	30.3109i	−0.554231	+	0.959955i	0.443732	+	0.896159i	$0.353654\pi$
$997$	−17.5000	+	30.3109i	−0.554231	+	0.959955i	−0.997963	+	0.0637961i	$0.979679\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4212.2.i.j.2809.1		2
3.2	odd	2		4212.2.i.c.2809.1		2
9.2	odd	6		1404.2.a.d.1.1	yes	1
9.4	even	3	inner	4212.2.i.j.1405.1		2
9.5	odd	6		4212.2.i.c.1405.1		2
9.7	even	3		1404.2.a.a.1.1	✓	1
36.7	odd	6		5616.2.a.c.1.1		1
36.11	even	6		5616.2.a.be.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1404.2.a.a.1.1	✓	1	9.7	even	3
1404.2.a.d.1.1	yes	1	9.2	odd	6
4212.2.i.c.1405.1		2	9.5	odd	6
4212.2.i.c.2809.1		2	3.2	odd	2
4212.2.i.j.1405.1		2	9.4	even	3	inner
4212.2.i.j.2809.1		2	1.1	even	1	trivial
5616.2.a.c.1.1		1	36.7	odd	6
5616.2.a.be.1.1		1	36.11	even	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 4212 = 2^{2} \cdot 3^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4212.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.50000 + 2.59808i) q^{5} +(-1.00000 + 1.73205i) q^{7} +(-0.500000 - 0.866025i) q^{13} +2.00000 q^{19} +(3.00000 + 5.19615i) q^{23} +(-2.00000 + 3.46410i) q^{25} +(-3.00000 + 5.19615i) q^{29} +(-1.00000 - 1.73205i) q^{31} -6.00000 q^{35} +2.00000 q^{37} +(-3.00000 - 5.19615i) q^{41} +(-2.50000 + 4.33013i) q^{43} +(-1.50000 + 2.59808i) q^{47} +(1.50000 + 2.59808i) q^{49} +12.0000 q^{53} +(1.50000 + 2.59808i) q^{59} +(-5.50000 + 9.52628i) q^{61} +(1.50000 - 2.59808i) q^{65} +(-4.00000 - 6.92820i) q^{67} +9.00000 q^{71} -4.00000 q^{73} +(-4.00000 + 6.92820i) q^{79} +(-4.50000 + 7.79423i) q^{83} -3.00000 q^{89} +2.00000 q^{91} +(3.00000 + 5.19615i) q^{95} +(-4.00000 + 6.92820i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{5} - 2 q^{7} - q^{13} + 4 q^{19} + 6 q^{23} - 4 q^{25} - 6 q^{29} - 2 q^{31} - 12 q^{35} + 4 q^{37} - 6 q^{41} - 5 q^{43} - 3 q^{47} + 3 q^{49} + 24 q^{53} + 3 q^{59} - 11 q^{61} + 3 q^{65}+ \cdots - 8 q^{97}+O(q^{100}) \) 2 * q + 3 * q^5 - 2 * q^7 - q^13 + 4 * q^19 + 6 * q^23 - 4 * q^25 - 6 * q^29 - 2 * q^31 - 12 * q^35 + 4 * q^37 - 6 * q^41 - 5 * q^43 - 3 * q^47 + 3 * q^49 + 24 * q^53 + 3 * q^59 - 11 * q^61 + 3 * q^65 - 8 * q^67 + 18 * q^71 - 8 * q^73 - 8 * q^79 - 9 * q^83 - 6 * q^89 + 4 * q^91 + 6 * q^95 - 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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